We study a category (Formula presented.) of (Formula presented.) -graded maximal Cohen-Macaulay (MCM) modules over the (Formula presented.) curve singularity and demonstrate that it has infinite type (Formula presented.) cluster combinatorics. In particular, we show that this Frobenius category (or a suitable subcategory) is stably equivalent to the infinite type (Formula presented.) cluster categories of Holm–Jørgensen, Fisher and Paquette–Yıldırım. As a consequence, (Formula presented.) has cluster tilting subcategories modelled by certain triangulations of the (completed) (Formula presented.) -gon. We use the Frobenius structure to extend this further to consider maximal almost rigid subcategories, and show that these subcategories and their mutations exhibit the combinatorics of the completed (Formula presented.) -gon.